Yes, no. But let me explain.
When I am asking this question, of course I am not just asking whether the man is meeting the everyday definition of the term that is nowadays common usage, as the answer to that should be more than obvious. No, I am more interested in the fact that this question has taken on such an importance and why this may be so.
The recent debate on Trumps mental state started with the open letter to the New York Times signed by more than 35 prominent mental health professionals warning the public about what they perceived as clear signs of Trumps emotional inability to be president. Soon responses by other prominent professionals followed warning about a diagnosis from afar and more outspoken then that: "It is a stigmatizing insult to the mentally ill (who are mostly well behaved and well meaning) to be lumped with Mr. Trump (who is neither). Bad behavior is rarely a sign of mental illness, and the mentally ill behave badly only rarely. Psychiatric name-calling is a misguided way of countering Mr. Trump’s attack on democracy. He can, and should, be appropriately denounced for his ignorance, incompetence, impulsivity and pursuit of dictatorial powers." (Alan Frances, Letter to the NYT).
Alan Frances objection to a mental health diagnosis seems to mostly come from a concern on how to respond to Trump. What his objection clearly implies is that you have to get the first question correct (the diagnostic problem, on which I disagree with Alan Frances) in order to answer to the more pressing matter of how should we respond and react to such a man.
This problem "of how to deal with unusual behaviour" frankly is an old one in our field, and has been addressed prominently in a simple heuristic: "mad, sad or bad". By this simple rule we could organize every "out of the ordinary" behaviour into a one of these categories and it was clear how to respond and treat the person, once so done. The mad (like the psychotic or manic person) would be seen driven by mental states beyond his control, thus be excused from the general expectations of rationality and decent behaviour, they would be locked up. The sad (like the depressed and anxious) would receive our empathy and support to lift them up until feeling better. The bad, of course the firm consequences and punishments that bad behaviour deserves.
Yet while this is a useful rule of thumb, the relation of these three (mad, bad sad) to each other has not been reflected much in our field, as can been seen in the quote above by a professor in psychiatry and the implicit assumption that these are three wholly distinct categories. "Bad behaviour is rarely a sign of mental illness" Mr. Frances writes, true. But what is even more important here is the implicit admission that hides in the rarely, that there is some small overlap between the two categories. And if you have ever read the criteria for narcissistic or anti-social personality disorder it is clear that "bad" is part and parcel of these conditions.
So where in this classification do the personality disorders fall and by extension then what should our reaction be? Here it becomes even more messy as it is unclear where to put them, rightly so, because actually we as humans and our personality is more dimensional and complex. We are neither simply good or bad, sad or mad, we are all of it! That's one reason why the same person can often elicit very different reactions in us. So rather than putting people once and for all in a box of "mad", "bad" or "sad" (as a diagnosis seems to be doing!) I suggest we respond to the human in front of us, each moment!
In regards to Donald Trump this means if he shows bad behaviour, it actually does not matter whether while so behaving he meets the criteria of a narcissistic personality disorder or not, bad is bad. It is on these grounds that we should resist him. Deciding what is bad is not the core business of the mental health profession, and it would be a sad day when we would have come so far that we need experts to tell us if somebody is behaving bad. So, no it should not matter to us whether he has the diagnosis of narcissistic personality disorder or not.